Damage Detection Using Two-Stage Compressive Sensing

ABSTRACT

Described herein are Compressive Sensing algorithms developed for automated reduction of NDE/SHM data from pitch-catch ultrasonic guided waves as well as a methodology using Compressive Sensing at two stages in the data acquisition and analysis process to detect damage: (1) temporally undersampled sensor signals from (2) spatially undersampled sensor arrays, resulting in faster data acquisition and reduced data sets without any loss in damage detection ability.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH

This disclosure was made with government support under NASA ContractNos. 80NSSC19C0592 and 80NSSC20C0200. The government has certain rightsin the disclosure.

TECHNICAL FIELD

The subject matter disclosed herein is generally directed to CompressiveSensing algorithms developed for automated reduction of NDE/SHM datafrom pitch-catch ultrasonic guided waves as well as a methodology usingCompressive Sensing at two stages in the data acquisition and analysisprocess to detect damage: (1) temporally undersampled sensor signalsfrom (2) spatially undersampled sensor arrays, resulting in faster dataacquisition and reduced data sets without any loss in damage detectionability.

BACKGROUND

NASA is developing new vehicles for human space flight. The Orion crewmodule is a reusable capsule that provides a habitat for the crew andfacilitates exploration of the Moon, asteroids, and Mars. The SpaceLaunch System (SLS) is a heavy lift system and is part of NASA's deepspace exploration plans. The SLS will carry humans beyond low Earthorbit and will deliver elements of the Lunar Orbital Platform-Gateway.Many of these spacecraft are targeted for long-term use, which offerschallenges for inspection and maintenance. In orbit or on the Moon, theuse of traditional NDE is prohibitive because of location andinaccessibility, and infrequent inspection can lead to conservative,high-weight designs. NASA is seeking technologies to facilitateinspections on large complex structures and provide reliable assessmentsof structural health.

Structural health monitoring (SHM) can help overcome inspectiondifficulties and has shown good results on small structures. However,transition to large complex structures has been slow. Some reasons forthe slow adoption are difficulties with large sensor arrays, timelyanalysis of large data sets, and overall weight of the system. In orderto realize the benefits of SHM, there is a need to reduce the number ofsensors and minimize data acquisition processes while maintaining theability to accurately detect, locate, and characterize damage.

Accordingly, it is an object of the present disclosure to help reducesensor data acquisition and processing burdens, as well as be used inapplications such as the Combined Loads Test System (COLTS) facility atNASA Langley Research Center.

Citation or identification of any document in this application is not anadmission that such a document is available as prior art to the presentdisclosure.

SUMMARY

The above objectives are accomplished according to the presentdisclosure by providing a method for employing compressive sensing tosense damage in a structure. The method may include employing at leastone spatially undersampled sensor array, obtaining at least onetemporally undersampled sensor signal from the at least one spatiallyundersampled sensor array, employing comprehensive sensing toreconstruct data from the at least one temporally undersampled sensorsignal to generate a reconstructed signal, and generating a diagnosticimage of the structure based on the reconstructed data. Further, the atleast one temporally undersampled sensor signal may result from missingactuator-sensor paths in the sensor array. Again, the method may beemployed to diagnose a space structure. Still further, the methodemploys at least one algorithm processed by an ultrasound softwarepackage. Yet again, the method may reconstruct at least one sensorsignal via using the at least one temporally undersampled signal, anappropriate basis function, and a measurement matrix. Still yet, themethod may utilize a subroutine to determine a best basis function forthe at least one temporally undersampled signal. Moreover, subroutinemay incorporate a Gini Index to find a basis function that provides asparsest signal in a transform domain. Even further, the at least onetemporally undersampled signal may be an ultrasound signal. Still again,the algorithm may be a sparse recovery algorithm. Yet further, thealgorithm may be a

1-norm minimization.

In a further embodiment, a diagnostic method for employing compressivesensing is provided. The method may include receiving a sensor signalfrom a sensor network affixed to a structure that transmits at least onesignal to the structure and receives the at least one signal afterencountering the structure, when the sensor signal received from thesensor network is undersampled in time and/or undersampled in space, thediagnostic method may engage at least one preprocessing module to runonly once to generate at least one basis function for at least onecompressive sensing algorithm, engage at least one signal reconstructionmodule containing the at least one compressive sensing algorithm,wherein the at least one reconstruction module reconstructs at least onetemporally undersampled sensor signal, engage at least one secondreconstruction module containing the at least one compressive sensingalgorithm to reconstruct at least one spatially undersampled sensorsignal, and determine whether damage has occurred to the structure viareconstruction of at least one temporally undersampled sensor signaland/or at least one spatially undersampled sensor signal. Further, thediagnostic method may be employed with at least one structural healthmonitoring system. Again, the diagnostic method may engage at least twopreprocessing modules. Yet still, the at least one signal may be anultrasound signal. Furthermore, the method may reconstruct at least onereconstructed sensor signal via using the at least one signal, anappropriate basis function, and a measurement matrix. Still yet, themethod may utilize a subroutine to determine a best basis function forthe at least one signal. Still further yet, the subroutine mayincorporate a Gini Index to find a basis function that provides asparsest signal in a transform domain. Moreover, the at least onecompressive sensing algorithm may be a sparse recovery algorithm. Stillyet, the sparse recovery algorithm may be a

1-norm minimization.

These and other aspects, objects, features, and advantages of theexample embodiments will become apparent to those having ordinary skillin the art upon consideration of the following detailed description ofexample embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

An understanding of the features and advantages of the presentdisclosure will be obtained by reference to the following detaileddescription that sets forth illustrative embodiments, in which theprinciples of the disclosure may be utilized, and the accompanyingdrawings of which:

FIG. 1 shows damage detection using two-stage compressive sensing.

FIG. 2 shows on embodiment of an architecture design of the currentdisclosure.

FIG. 3 shows one embodiment of a finding the best basis function of thecurrent disclosure.

FIG. 4 shows a reconstruction of sensor signals for the currentdisclosure.

FIG. 5 shows Stage 1 takes the basis function, Ψ₁, and the undersampleddata as inputs, and generates the necessary data for Stage 2.

FIG. 6 shows an example signal given byf(t)=sin(2π(98000)t)+sin(2π(122000)t).

FIG. 7 shows amplitude versus time for 64 random sample points.

FIG. 8 shows an exact reconstruction of a signal.

FIG. 9 shows a comparison of a reconstructed and fully-sampled signal.

FIG. 10 shows an ultrasound signal from a composite plate.

FIG. 11 shows amplitude versus time for 240 random sample points.

FIG. 12 shows close reconstruction of a sensor signal.

FIG. 13 shows a comparison of a reconstructed and fully-sampled signal.

FIG. 14 shows a baseline subtraction to find scatter signal.

FIG. 15 damage acts an emitter of scatter signals.

FIG. 16 the output of Stage 1 is the scatter signals from the paths thatwere sampled.

FIG. 17 shows reconstruction of sensor array points.

FIG. 18 shows feature extraction.

FIG. 19 shows discretized structure and a single actuator-sensor path.

FIG. 20 shows scatter signal expanded in basis, Ψ.

FIG. 21 shows generating the Stage 2 basis function.

FIG. 22 shows Stage 2 takes the basis function, Ψ₂, and the scattersignals as inputs, and generates the Damage Indices, which are then usedto create a diagnostic image.

FIG. 23 shows software architecture design.

FIG. 24 show an experimental setup of the current disclosure.

FIG. 25 shows damage simulators attached to composite structure.

FIG. 26 shows diagnostic image from fully-sampled data.

FIG. 27 shows a comparison of diagnostic images from fully-sampled andundersampled data.

FIG. 28 shows a diagnostic images as a function of sampled points andpaths.

FIG. 29 shows graphical user interface for Stage 1 SignalReconstruction.

FIG. 30 shows an example signal reconstruction.

FIG. 31 shows an example of correlation coefficients.

FIG. 32 shows a flowchart of internal algorithms, as well as detailedinputs and outputs of each of the modules of the current disclosure.

FIG. 33 shows one embodiment of a refined and generalized softwarearchitecture design of the current disclosure.

FIG. 34 shows POD curves for traditional sensing shift to the right whenfewer measurements are used.

FIG. 35 shows POD curves for compressive sensing move back toward thefully sampled case.

FIG. 36 shows quantified savings versus CR and POD.

FIG. 37A shows a graph of A scan data.

FIG. 37B shows C-scan data and an image thereof.

FIG. 38 shows a composite plate geometry and dimensions.

FIG. 39 shows a C-scan through-transmission image after LBI.

FIG. 40 shows an image using Z-scan pulse-echo data at the bondline.

FIG. 41 shows C-scan images as a function of A-scan and C-scancompression ratios.

FIG. 42 shows a graph of POD curves for traditional sensing shift to theright when fewer measurements are used.

FIG. 43 shows a graph of POD curves for compressive sensing move backtoward the fully sampled case.

FIG. 44 shows a graph of example POD curves for undersampled data beforeCS reconstruction.

FIG. 45 shows a graph of example POD curves for undersampled data afterCS reconstruction.

FIG. 46 shows a graph of example POD curves for undersampled data afterCS reconstruction.

FIG. 47 shows a graph of quantified savings vs. CR and POD.

FIG. 48 shows a graph of data storage space vs. CR and POD.

The figures herein are for illustrative purposes only and are notnecessarily drawn to scale.

DETAILED DESCRIPTION OF THE EXAMPLE EMBODIMENTS

Before the present disclosure is described in greater detail, it is tobe understood that this disclosure is not limited to particularembodiments described, and as such may, of course, vary. It is also tobe understood that the terminology used herein is for the purpose ofdescribing particular embodiments only, and is not intended to belimiting.

Unless specifically stated, terms and phrases used in this document, andvariations thereof, unless otherwise expressly stated, should beconstrued as open ended as opposed to limiting. Likewise, a group ofitems linked with the conjunction “and” should not be read as requiringthat each and every one of those items be present in the grouping, butrather should be read as “and/or” unless expressly stated otherwise.Similarly, a group of items linked with the conjunction “or” should notbe read as requiring mutual exclusivity among that group, but rathershould also be read as “and/or” unless expressly stated otherwise.

Furthermore, although items, elements or components of the disclosuremay be described or claimed in the singular, the plural is contemplatedto be within the scope thereof unless limitation to the singular isexplicitly stated. The presence of broadening words and phrases such as“one or more,” “at least,” “but not limited to” or other like phrases insome instances shall not be read to mean that the narrower case isintended or required in instances where such broadening phrases may beabsent.

Unless defined otherwise, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this disclosure belongs. Although any methods andmaterials similar or equivalent to those described herein can also beused in the practice or testing of the present disclosure, the preferredmethods and materials are now described.

All publications and patents cited in this specification are cited todisclose and describe the methods and/or materials in connection withwhich the publications are cited. All such publications and patents areherein incorporated by references as if each individual publication orpatent were specifically and individually indicated to be incorporatedby reference. Such incorporation by reference is expressly limited tothe methods and/or materials described in the cited publications andpatents and does not extend to any lexicographical definitions from thecited publications and patents. Any lexicographical definition in thepublications and patents cited that is not also expressly repeated inthe instant application should not be treated as such and should not beread as defining any terms appearing in the accompanying claims. Thecitation of any publication is for its disclosure prior to the filingdate and should not be construed as an admission that the presentdisclosure is not entitled to antedate such publication by virtue ofprior disclosure. Further, the dates of publication provided could bedifferent from the actual publication dates that may need to beindependently confirmed.

As will be apparent to those of skill in the art upon reading thisdisclosure, each of the individual embodiments described and illustratedherein has discrete components and features which may be readilyseparated from or combined with the features of any of the other severalembodiments without departing from the scope or spirit of the presentdisclosure. Any recited method can be carried out in the order of eventsrecited or in any other order that is logically possible.

Where a range is expressed, a further embodiment includes from the oneparticular value and/or to the other particular value. The recitation ofnumerical ranges by endpoints includes all numbers and fractionssubsumed within the respective ranges, as well as the recited endpoints.Where a range of values is provided, it is understood that eachintervening value, to the tenth of the unit of the lower limit unlessthe context clearly dictates otherwise, between the upper and lowerlimit of that range and any other stated or intervening value in thatstated range, is encompassed within the disclosure. The upper and lowerlimits of these smaller ranges may independently be included in thesmaller ranges and are also encompassed within the disclosure, subjectto any specifically excluded limit in the stated range. Where the statedrange includes one or both of the limits, ranges excluding either orboth of those included limits are also included in the disclosure. Forexample, where the stated range includes one or both of the limits,ranges excluding either or both of those included limits are alsoincluded in the disclosure, e.g. the phrase “x to y” includes the rangefrom ‘x’ to ‘y’ as well as the range greater than ‘x’ and less than ‘y’.The range can also be expressed as an upper limit, e.g. ‘about x, y, z,or less’ and should be interpreted to include the specific ranges of‘about x’, ‘about y’, and ‘about z’ as well as the ranges of ‘less thanx’, less than y′, and ‘less than z’. Likewise, the phrase ‘about x, y,z, or greater’ should be interpreted to include the specific ranges of‘about x’, ‘about y’, and ‘about z’ as well as the ranges of ‘greaterthan x’, greater than y′, and ‘greater than z’. In addition, the phrase“about ‘x’ to ‘y’”, where ‘x’ and ‘y’ are numerical values, includes“about ‘x’ to about ‘y’”.

It should be noted that ratios, concentrations, amounts, and othernumerical data can be expressed herein in a range format. It will befurther understood that the endpoints of each of the ranges aresignificant both in relation to the other endpoint, and independently ofthe other endpoint. It is also understood that there are a number ofvalues disclosed herein, and that each value is also herein disclosed as“about” that particular value in addition to the value itself. Forexample, if the value “10” is disclosed, then “about 10” is alsodisclosed. Ranges can be expressed herein as from “about” one particularvalue, and/or to “about” another particular value. Similarly, whenvalues are expressed as approximations, by use of the antecedent“about,” it will be understood that the particular value forms a furtheraspect. For example, if the value “about 10” is disclosed, then “10” isalso disclosed.

It is to be understood that such a range format is used for convenienceand brevity, and thus, should be interpreted in a flexible manner toinclude not only the numerical values explicitly recited as the limitsof the range, but also to include all the individual numerical values orsub-ranges encompassed within that range as if each numerical value andsub-range is explicitly recited. To illustrate, a numerical range of“about 0.1% to 5%” should be interpreted to include not only theexplicitly recited values of about 0.1% to about 5%, but also includeindividual values (e.g., about 1%, about 2%, about 3%, and about 4%) andthe sub-ranges (e.g., about 0.5% to about 1.1%; about 5% to about 2.4%;about 0.5% to about 3.2%, and about 0.5% to about 4.4%, and otherpossible sub-ranges) within the indicated range.

As used herein, the singular forms “a”, “an”, and “the” include bothsingular and plural referents unless the context clearly dictatesotherwise.

As used herein, “about,” “approximately,” “substantially,” and the like,when used in connection with a measurable variable such as a parameter,an amount, a temporal duration, and the like, are meant to encompassvariations of and from the specified value including those withinexperimental error (which can be determined by e.g. given data set, artaccepted standard, and/or with e.g. a given confidence interval (e.g.90%, 95%, or more confidence interval from the mean), such as variationsof +/−10% or less, +/−5% or less, +/−1% or less, and +/−0.1% or less ofand from the specified value, insofar such variations are appropriate toperform in the disclosure. As used herein, the terms “about,”“approximate,” “at or about,” and “substantially” can mean that theamount or value in question can be the exact value or a value thatprovides equivalent results or effects as recited in the claims ortaught herein. That is, it is understood that amounts, sizes,formulations, parameters, and other quantities and characteristics arenot and need not be exact, but may be approximate and/or larger orsmaller, as desired, reflecting tolerances, conversion factors, roundingoff, measurement error and the like, and other factors known to those ofskill in the art such that equivalent results or effects are obtained.In some circumstances, the value that provides equivalent results oreffects cannot be reasonably determined. In general, an amount, size,formulation, parameter or other quantity or characteristic is “about,”“approximate,” or “at or about” whether or not expressly stated to besuch. It is understood that where “about,” “approximate,” or “at orabout” is used before a quantitative value, the parameter also includesthe specific quantitative value itself, unless specifically statedotherwise.

The term “optional” or “optionally” means that the subsequent describedevent, circumstance or substituent may or may not occur, and that thedescription includes instances where the event or circumstance occursand instances where it does not.

Various embodiments are described hereinafter. It should be noted thatthe specific embodiments are not intended as an exhaustive descriptionor as a limitation to the broader aspects discussed herein. One aspectdescribed in conjunction with a particular embodiment is not necessarilylimited to that embodiment and can be practiced with any otherembodiment(s). Reference throughout this specification to “oneembodiment”, “an embodiment,” “an example embodiment,” means that aparticular feature, structure or characteristic described in connectionwith the embodiment is included in at least one embodiment of thepresent disclosure. Thus, appearances of the phrases “in oneembodiment,” “in an embodiment,” or “an example embodiment” in variousplaces throughout this specification are not necessarily all referringto the same embodiment, but may. Furthermore, the particular features,structures or characteristics may be combined in any suitable manner, aswould be apparent to a person skilled in the art from this disclosure,in one or more embodiments. Furthermore, while some embodimentsdescribed herein include some but not other features included in otherembodiments, combinations of features of different embodiments are meantto be within the scope of the disclosure. For example, in the appendedclaims, any of the claimed embodiments can be used in any combination.

All patents, patent applications, published applications, andpublications, databases, websites and other published materials citedherein are hereby incorporated by reference to the same extent as thougheach individual publication, published patent document, or patentapplication was specifically and individually indicated as beingincorporated by reference.

Compressive Sensing has been shown to greatly reduce data acquisitionand processing burdens by providing mathematical guarantees for accuratesignal recovery from far fewer samples than conventionally needed.Compressive Sensing algorithms were developed for automated reduction ofNDE/SHM data from pitch-catch ultrasonic guided waves. The methodologyuses Compressive Sensing at two stages in the data acquisition andanalysis process to detect damage. The two stages are: (1) temporallyundersampled sensor signals from (2) spatially undersampled sensorarrays, resulting in faster data acquisition and reduced data setswithout any loss in damage detection ability.

The purpose of the innovation is to reduce data acquisition processesand data storage burdens for NDE/SHM systems while maintaining theability to accurately detect, locate, and characterize structuraldamage. The methodology uses Compressive Sensing to reconstruct data attwo stages in the data acquisition and analysis process to detect damageand generate a diagnostic image of the structure.

Data may be reconstructed at two stages: (1) temporally undersampledsensor signals from (2) spatially undersampled sensor arrays. Aconceptual diagram of the methodology is shown at FIG. 1 . FIG. 1 at thetop row depicts Stage 1, where temporally undersampled signals arereconstructed using Compressive Sensing. The reconstructed signals arethen analyzed and features are extracted to provide inputs into Stage 2,which is depicted in the bottom row. In Stage 2, spatially undersampledsensor data (from missing actuator-sensor paths) are reconstructed togenerate a diagnostic image of the structure.

FIG. 2 shows the methodology of FIG. 1 broken down into four modules asshown in the diagram. The preprocessing modules are intended to be runonly once in order to generate the basis functions required for theCompressive Sensing algorithms that are housed in the Stage 1 and Stage2 modules. The Stage 1 Signal Reconstruction module contains CompressiveSensing algorithms to enable reconstruction of temporally undersampledsensor signals. The Stage 2 Path Reconstruction module containsCompressive Sensing algorithms for reconstruction of spatiallyundersampled actuator-sensor paths.

It is anticipated that the first application of the technology will bethe integration into NASA's inspection tools for large complex spacestructures made with composites or thin metals, such as the Orion crewmodule, Space Launch System, and the Lunar Outpost Platform-Gateway. AsNASA continues to direct efforts into deep space flight, smartstructures that are instrumented with structural health monitoring (SHM)systems will be needed to provide real time mission critical informationof the structure's status. In order for these SHM systems to be viable,the total number of sensors, total weight, and data acquisitionrequirements must be minimized, and the Compressive Sensing methodologywill be critical in achieving this.

Potential commercial customers include Blue Origin and SpaceX, whichbuild large, reusable space launch vehicles. Other non-NASA applicationsand industries include aerospace (aircraft wings and fuselage), marine(ship hulls), wind energy (rotor blades), transportation/railways, civilinfrastructure (buildings and bridges), oil and gas (pipelines), etc.Generally, any industry that uses large structures that require frequentinspection will benefit from the use of the Compressive Sensingtechnology. Similar benefits can also be realized in the emergingwearable sensors market in the healthcare industry.

Compressive Sensing (CS) has been shown to greatly reduce dataacquisition and processing burdens by providing mathematical guaranteesfor accurate signal recovery from far fewer samples than conventionallyneeded. In Phase I of this project, algorithms for an ultrasoundsoftware package were developed to detect damage in structures using CSat two stages in the data acquisition and analysis process as shown inFIG. 1 . The two stages are: (1) temporally undersampled sensor signalsfrom (2) spatially undersampled sensor arrays, resulting in faster dataacquisition and reduced data sets without any loss in damage detectionability.

The ultimate goal of this project is to reduce data acquisitionrequirements (energy consumption, number of sensors, data collection andstorage, and total system weight) of NDE/SHM systems. Advent teamed withthe University of South Carolina (USC) to design and develop thesoftware package for automated reduction of NDE/SHM data frompitch-catch ultrasonic guided waves.

The specific objectives in the Phase I period were to:

Design a hierarchical modular software architecture for CompressiveSensing of ultrasonic guided waves that is flexible and scalable,allowing it to be easily extended to additional NDE/SHM techniques.

Develop a superior version of USC's feature extraction toolbox (ZIGANAL)that is expanded and enhanced with CS-specific algorithms.

Define and encode orthonormal basis functions to enable robustultrasonic wave signal reconstruction.

Construct a Damage Index formulation that is robust and consistent withCompressive Sensing theory.

Automate the entire process from data acquisition to display of detecteddamage.

Conduct extensive functionality testing on experimental and simulateddata to demonstrate feasibility.

Software Architecture Design

The software architecture was designed to be flexible and reconfigurableto facilitate rapid module development, simplify module verification andvalidation, and allow for efficient upgrades to the system. Theindividual modules were developed based on compressive sensing (CS)theory.

Compressive Sensing Background

CS theory states that one can recover certain signals and images fromfar fewer samples or measurements than traditional methods (e.g.Nyquist-Shannon). To make this possible, CS relies on two principles:sparsity, which pertains to the signals of interest, and incoherence,which pertains to the sensing modality.

As an example, consider a vector f∈

^(n) (such as a sensor signal or n-pixel image), which is expanded in anorthonormal basis (such as a wavelet basis) Ψ=[ψ₁ ψ₂ . . . ψ_(n)] asfollows:

f(t)=Σ_(i=1) ^(n) x _(i)ψ_(i)(t) or f=Ψx

where x is the coefficient sequence of f. x_(i)=(f, ψ_(i)) or x=Ψ^(T)fand has k nonzero elements (k<<n, k-sparse).

Suppose f has been undersampled during data acquisition, such that y=Φf,where y∈

^(m) (m<n) and Φ is the measurement matrix (Φ∈

^(m×n)). The vector y can be written as

y=Φf=ΦΨx=Θx

where Θ is an m×n transfer matrix with much fewer rows than columns.Construction of x from y is an underdetermined ill-posed inverse problembecause the dimension of y is much small than that of x. However, it hasbeen proven that x and f can be uniquely reconstructed with overwhelmingprobability if the signal f is sparse and the transfer matrix Θ meetsthe so-called restricted isometry property (RIP). That is, there existsan isometric constant δ_(k) for the matrix Θ where δ_(k) is defined asthe smallest number which holds for all k-sparse vectors, x, such that:

${1 - \delta_{k}} \leq \frac{{{\Theta x}}_{2}^{2}}{{x}_{2}^{2}} \leq {1 + \delta_{k}}$

This property essentially requires that every set of columns withcardinality less than k are approximately orthonormal. If the columns ofthe transfer matrix Θ are orthogonal, then x can be exactly constructedfrom y. Specifically, incoherence between the sensing (sampling) matrix,Φ, and the dictionary (matrix of representation bases), Ψ, is arequirement to satisfy the RIP and achieve accurate reconstruction withhigh probability. Random matrices are largely incoherent with any fixedbasis Ψ, and for that reason, random sampling was used in the softwaredesign.

The signal is then reconstructed through a sparse recovery algorithm,such as

1-norm minimization.

The software architecture consists of four separate modules, as shown inFIG. 2 , consisting of two preprocessing modules and two reconstructionmodules.

The preprocessing modules are intended to be run only once to generatethe basis functions required for the CS algorithms housed in the Stage 1and Stage 2 modules. The Stage 1 Signal Reconstruction module containsCS algorithms to enable Reconstruction of Temporally Undersampled SensorSignals. The Stage 2 Path Reconstruction module contains CS algorithmsfor Reconstruction of Spatially Undersampled Sensor Arrays. Thedevelopment of the Preprocessing for Stage 1 module is discussed infraalong with development of the Stage 1 Signal Reconstruction module.Development of the Preprocessing for Stage 2 module, as well as theStage 2 Path Reconstruction module, are also discussed infra.

Automated Feature Extraction Module

The subcontractor, USC, built upon previous work to enhance the existingfeature extraction toolbox named ZIGANAL. CS tools and algorithms wereadded to analyze sparse signals. Specifically, a subroutine wasdeveloped to find the best basis function to use for a given signal.

The purpose of Stage 1 is to fully reconstruct a sensor signal given theundersampled signal, an appropriate basis function, and the measurementmatrix:

But for accurate signal reconstruction, an appropriate basis functionmust first be found. The Preprocessing for Stage 1 module utilizes asubroutine to find the best basis function for a given signal. Thissubroutine incorporates the use of the Gini Index to find the basisfunction that results in the sparsest signal in the transform domain. Anassortment of potential basis functions (FIG. 3 ) are tested andanalyzed to determine the basis function that results in the highestcorrelation coefficient between the reconstructed signal and thefully-sampled signal.

The ultrasound signals being studied in Phase I of this project weresinusoidal in nature, and therefore, the Preprocessing for Stage 1module consistently selected the Fast Fourier Transform and the DiscreteCosine Transform as the best basis functions for the signals.

Module to Reconstruct Temporally Undersampled Sensor Signals

The Stage 1 Signal Reconstruction module and submodules were developedin this task. The purpose of this module is to reconstruct undersampledsensor signals using CS theory (FIG. 4 ), and to provide input intoStage 2 Path Reconstruction (FIG. 5 ).

The first part of this module uses CS theory with

1-norm minimization as the sparse recovery algorithm to reconstruct theundersampled sensor signals. Two example signal reconstructions usingthe module are shown below.

Example 1

The first example is the digital signal shown in FIG. 6 . The signal isa combination of two sinusoids given by the following equation:

f(t)=sin(2π(98000)t)+sin(2π(122000)t)

The sampling frequency is 256 kHz and the number of sample points in256.

Using the subroutine developed herein to find the best basis function,it was found the best basis function to use for this particular signalwas Ψ=Fast Fourier Transform. A random measurement matrix was used toobtain 64 random sample points from the signal as shown in FIG. 7 .

Given only the undersampled signal (64 points), basis function (FastFourier Transform), and the random measurement matrix, the sparserecovery algorithm (

1-norm minimization) was able to reconstruct the signal exactly as shownin FIG. 8 .

One hundred and thirty cases were run with a varying number of randomsample points to compare the efficacy of the reconstruction. Thecorrelation coefficient of the reconstructed and fully-sampled signalvs. number of random samples is shown in FIG. 9 . In this plot, acorrelation coefficient of 1 indicates an exact reconstruction, andanything above 0.8 is consider a very close reconstruction. As can beseen, using between 40 and 60 random samples results in goodreconstruction, and anything above 60 random samples results in exactreconstruction.

Example 2

The second example is a real pitch-catch ultrasound signal (FIG. 10 )from a composite plate. The sampling frequency is 1 MHz and the numberof sample points is 800.

Using the subroutine developed herein to find the best basis function,it was found the best basis function to use for this particular signalwas Ψ=Discrete Cosine Transform. A random measurement matrix was used toobtain 240 random sample points from the signal as shown in FIG. 11 .

Given only the undersampled signal (240 points), basis function(Discrete Cosine Transform), and the random measurement matrix, thesparse recovery algorithm (

1-norm minimization) was able to perform a close reconstruction of thesignal as shown in FIG. 12 .

475 cases were run with varying number of random sample points tocompare the efficacy of the reconstruction. The correlation coefficientof the reconstructed and fully-sampled signal vs. number of randomsamples is shown in FIG. 13 . As mentioned previously, a correlationcoefficient of 1 indicates an exact reconstruction, and anything above0.8 is consider a very close reconstruction. As can be seen, usingbetween ˜175 and ˜325 random samples results in good reconstruction, andanything above ˜325 random samples results in exact reconstruction.

After the signals have been reconstructed, they are ready to beanalyzed. In addition to the CS submodules above, a “baselinesubtraction” submodule was also added. The recovered signals arecompared to previously collected baseline signals from the undamagedstructure to obtain scatter signals, which are just the differencebetween the current sensor data and the baseline sensor data (FIG. 14 ).

The scatter signals contain information about any existing damage in thestructure. When a propagating wave encounters damage, the wave isreflected, refracted, and/or diffracted, or some combination of allthree, depending on the damage type. The scatter signals represent thereflected, refracted, and diffracted waves, which all emanate from thedamage(s). When viewing the scatter field, the damage acts as an emittersource of the scatter signals (FIG. 15 ).

The output of the Stage 1 Signal Reconstruction module is the scattersignals from all paths that were sampled (FIG. 16 ). These scattersignals are then used in the Stage 2 Path Reconstruction moduledescribed infra.

Module to Reconstruct Spatially Undersampled Sensor Arrays

The Preprocessing for Stage 2 and the Stage 2 Path Reconstructionmodules and submodules were developed in this task. The purpose of Stage2 is to reconstruct spatially undersampled sensor arrays using CS theory(FIG. 17 ). Specifically, scatter signals from undersampledactuator-sensor paths are reconstructed in order to generate adiagnostic image of the structure without losing accuracy if there aremissing actuator-sensor paths.

One method to generate a diagnostic image is to extract features (suchas time-of-flight, amplitude, energy) from the scatter signals (FIG. 18) of every actuator-sensor path and calculate a parameter called aDamage Index.

The Damage Indices from each actuator-sensor path can be averaged overthe entire structure by discretizing the structure into k grid points:

$D_{k} = {\frac{1}{N}{\sum}_{i = 1}^{N}{D_{i}\left( {{TOF}_{ik},A_{ik},E_{ik}} \right)}}$

where k is the grid point, i is the path number, and N is the totalnumber of actuator-sensor paths. The diagram in FIG. 19 shows thediscretized structure and a single actuator-sensor path. The time ittakes for a wave to propagate from an actuator, a, to a given gridpoint, k, and then from the grid point to a given sensor, s, is given by

$t_{ak} = {{\frac{\sqrt{\left( {x_{a} - x_{k}} \right)^{2} + \left( {y_{a} - y_{k}} \right)^{2}}}{v_{ak}}{and}t_{ks}} = \frac{\sqrt{\left( {x_{s} - x_{k}} \right)^{2} + \left( {y_{s} - y_{k}} \right)^{2}}}{v_{ks}}}$

respectively

where x_(a) and y_(a) are the actuator coordinates, x_(s) and y_(s) arethe sensor coordinates, x_(k) and y_(k) are the grid point coordinates,and v_(ak) and v_(ks) are the wave velocities, which will vary withdirection in anisotropic materials. The wave velocity profile inanisotropic materials is obtained from the time-of-arrival of the firstwave packets in the baseline signals. Therefore, in a given scattersignal, the time-of-arrival of a wave emanating from a given grid pointis TOF=t_(ak)+t_(ks). The resulting D_(k) represent the probability thatdamage exists at a particular grid point, k, and are used to generatethe diagnostic image.

Conversely, each scatter signal can be expanded in a basis, Ψ=[ψ₁ ψ₂ . .. ψ_(n)], with D_(k) as the coefficient sequence as shown in FIG. 20 .Note that this formulation is now consistent with CS theory.

The Preprocessing for Stage 2 module contains a subroutine toautomatically generate the Ψ array for a given structural application(FIG. 21 ).

With the generated Ψ₂ array, and the Damage Index relations constructedin a formulation consistent with CS theory, the next step is Stage 2Path Reconstruction. The Stage 2 Path Reconstruction module takes thebasis function, Ψ₂, and the scatter signals as inputs, and uses CStheory to generate the Damage Indices, which are then used to create adiagnostic image (FIG. 22 ).

The sparse recovery utilizes

1-norm minimization similar to that in Stage 1, but with addeddimensionality because y is a matrix instead of a vector, and Ψ₂ is a3-dimensional array instead of a 2-dimensional matrix.

The recovered Damage Index values, D_(k), are then mapped to theirrespective grid point coordinates on the structure, and a diagnosticimage is generated highlighting the most probable location and size ofany damage on the structure.

The complete software architecture, with inputs and outputs to eachmodule, is shown in FIG. 23 .

Functionality Testing/Demonstration

In this task, a composite structure was instrumented with sixteenpiezoelectric sensors and used to test the algorithms and entiresoftware package to demonstrate damage detectability using reduced setsof data. Damage detection results with various levels of downsamplingwere compared with that from the fully sampled case. In addition, agraphical user interface was developed to help guide and visualize thesignal reconstruction process.

Functionality Testing

The experimental setup is shown in FIG. 24 , which includes oscilloscope2402, function generator 2404, power amplifier 2406 and composite plate2408. The sixteen piezoelectric transducers 2410 were attached in asquare grid, and each are used as both an actuator and sensor totransmit and receive ultrasonic waves in a pitch-catch mode. A full setof healthy “baseline data” was collected from the 120 uniqueactuator-sensor paths, and the data and structural parameters were runthrough the Preprocessing for Stage 1 and Preprocessing for Stage 2modules to generate the basis functions for Stages 1 and 2.

Two damage simulators 2502 were then attached to the composite structureas shown in FIG. 25 . The damage simulators are steel cylinders bondedto the surface of the structure. This causes a change in the localeffective stiffness, which reflects, refracts, and diffracts propagatingultrasound waves, mimicking damage in the structure. Using damagesimulators has the advantage of being able to run many scenarios withdifferent damage sizes and locations, but all nondestructively.

With the damage simulators attached, a set of “current data” wascollected from the structure and run through Stage 1 and Stage 2 togenerate the diagnostic image from the fully-sampled signals (1000points) and paths (120 paths) as shown in FIG. 26 .

Next, 400 random signal points from 80 random actuator-sensor paths weresampled and run through Stages 1 and 2 to compare the diagnostic imagefrom the undersampled data to that from the fully-sampled data (FIG. 27). As can be seen, there is some noise in the image from theundersampled data, but is still clear where the damage is located.

A parametric study was conducted by varying the number of sampledsignals points and number of sampled paths. The generated diagnosticimages as a function of sampled points and paths are plotted in FIG. 28. The diagnostic images exhibit more noise as fewer sampled signalspoints and fewer sampled paths are used in the analysis. The red curvedarrow delineates good diagnostic images from those that are too noisy tobe used reliably. The images to the right and above the arrow areconsidered to be good, and the images to the left and below the arroware considered to be too noisy.

Graphical User Interface

To help guide and visualize the Stage 1 Signals Reconstruction process,a graphical user interface (GUI) was developed (FIG. 29 ). The GUIcontains multiple analysis options for a variety of basis functions, andthe user can visualize and quantify signal comparisons via crosscorrelation.

An example reconstruction is shown in FIG. 30 , where a 10,000 pointultrasound signal has been randomly downsampled to 1000 points(compression ratio=10%) and then reconstructed using the Discrete CosineTransform as the basis function. The fully-sampled signal and thereconstructed signal are shown in the lower window.

The “Correlation Coefficient” tab displays a plot of the correlationcoefficients to quantify reconstruction ability vs. sample points forall selected basis functions (FIG. 31 ).

This Phase I effort focused on developing Compressive Sensing algorithmsand modules for automated reduction of NDE/SHM data from pitch-catchultrasonic guided waves. The methodology uses Compressive Sensing at twostages in the data acquisition and analysis process to detect damage.The two stages are: (1) temporally undersampled sensor signals from (2)spatially undersampled sensor arrays, resulting in faster dataacquisition and reduced data sets without any loss in damage detectionability.

Compressive Sensing algorithms and prototype software modules weredeveloped for both stages to reconstruct undersampled sensor signals andreconstruct Damage Indices from undersampled actuator-sensor paths.Functionality testing and technical feasibility was successfullydemonstrated on a composite structure using an array of piezoelectricsensors to generate diagnostic images from various levels ofundersampled data.

Future Work and Potential Applications

While the Phase I focus was on reducing the data requirements forpitch-catch ultrasonic guided waves, the open system architecture designof the software is modular and scalable, allowing it to be extended toother NDE/SHM ultrasound techniques, such as pulse-echo, acousticemission, shear wave, and impact detection. The focus of the Phase IIeffort will be to expand the modules to include other NDE/SHM ultrasoundinspection and diagnostic techniques, develop interfaces to prognosticmodels, conduct comprehensive software verification and validationtesting, and develop hardware specifications and designs to takeadvantage of the Compressive Sensing software to minimize weight andreduce the number sensors needed for accurate damage detection. The goalwill be to turn this concept into a commercially viable product thatwill be available for widespread trial and adaptation at the end of thePhase II project.

Specifically, in Phase II, Advent will:

Develop and integrate modules for pulse-echo, acoustic emission, shearwave, and impact detection.

Collaborate with NASA and OEMs to evaluate and refine the softwarepackage. Incorporate feedback on the operation and usability from endusers.

Conduct comprehensive verification and validation of the technologyacross a variety of large complex space structures.

Develop hardware specifications to take advantage of the CompressiveSensing software to reduce data acquisition requirements compared toconventional systems.

Conduct study to quantify the reduction in energy consumption, number ofsensors, data acquisition and storage requirements, and total systemweight resulting from the use of Compressive Sensing.

In collaboration with NASA and industrial partners, develop animplementation and technology transfer plan for incorporating the systeminto their sustained maintenance planning.

In the Phase III effort, Advent will assist in transitioning the systemto NASA and provide support in system integration and qualificationtesting of the software technology. The technology can potentially betested and used in the Combined Loads Test System (COLTS) facility atNASA Langley Research Center to help reduce sensor data acquisition andprocessing burdens.

It is anticipated that the first application of the technology will bethe integration into NASA's inspection tools for large complex spacestructures made with composites or thin metals, such as the Orion crewmodule, Space Launch System, and the Lunar Outpost Platform-Gateway. AsNASA continues to direct efforts into deep space flight, smartstructures that are instrumented with SHM systems will be needed toprovide real time mission critical information of the structure'sstatus. In order for these SHM systems to be viable, the total number ofsensors, total weight, and data acquisition requirements must beminimized, and Advent's Compressive Sensing software will be critical inachieving this.

Internal algorithms, as well as detailed inputs and outputs of each ofthe modules, are shown in the flowchart in FIG. 32 . In the flowchart,the white parallelograms are input data/parameters. The greenparallelograms are internal parameters that get passed betweenfunctions, and the functions are the blue boxes.

Novel features of the innovation include: (1) the ability to generatebasis functions needed for Compressive Sensing, (2) the ability toreconstruct temporally undersampled sensor signals, (3) the ability toreconstruct parameters from spatially undersampled actuator-sensorpaths, and (4) the ability to generate diagnostic images from thereconstructed data. The advantages of the innovation are the reductionin data acquisition processes and storage, and the potential to reducethe number of required sensors and total weight of NDE/SHM systems.

Refined and Expanded Software Architecture

The focus of the Phase I effort was on developing an open systemarchitecture (OSA) software design for reconstruction of undersampledNDE/SHM data from pitch-catch ultrasonic guided waves. Advent designedthe software architecture to be flexible and reconfigurable tofacilitate rapid module development, simplify module verification andvalidation, and allow for efficient upgrades to the system. While thePhase I focus was on reducing the data requirements for pitch-catchultrasonic guided waves, the OSA software design allows it to beextended to other NDE/SHM techniques. In Phase II, Advent is buildingupon the modular software architecture design and expanding it toinclude other NDE/SHM techniques. The existing code is being refined andstreamlined to optimize Stage 1 and Stage 2 computation of large datasets, and additional modules are being added to include datareconstruction capabilities for pulse-echo (A-scan), C-scan, acousticemission, impact data, thermography, and Terahertz scanning data.

The software architecture developed in Phase I is comprised of fourseparate modules, consisting of two preprocessing modules and tworeconstruction modules, see FIG. 2 .

Stage 1 uses CS to reconstruct undersampled pitch-catch ultrasoundsignals (undersampled in time), and Stage 2 uses CS to reconstruct datafrom missing actuator-sensor paths (undersampled in space). But not allNDE/SHM methods will generate data that can be undersampled in both timeand space. Therefore, to generalize the architecture to accommodateother types of NDE/SHM methods, the stages have been redefined to beStage 1 Temporal Reconstruction and Stage 2 Spatial Reconstruction.Also, the NDE/SHM dataflow has been designed to distinguish betweendatasets that have been undersampled in both time and space from thosethat have been undersampled in either time or space, see FIG. 33 Forexample, data that has been undersampled in both space and time (e.g.the pitch-catch data from Phase I) will flow through both Stage 1 andStage 2 as shown by the blue arrows in FIG. 33 . Whereas, NDE data thathas been undersampled in time only (e.g. pulse-echo/A-scan or acousticemission data) will just flow through Stage 1 to reconstruct the signalsas shown by the red arrows. Similarly, data that has been undersampledin space only (e.g. C-scan data) will bypass Stage 1 and go directly toStage 2 to reconstruct the data for imaging as shown by the greenarrows. The Preprocessing modules for Stage 1 and Stage 2 are intendedto be run only once for each application. Their purpose is to determinethe best basis functions to use for a particular type of NDE/SHM data toaccurately reconstruct the data in time (Stage 1) and/or space (Stage2).

Methodology to Determine Probability-of-Detection as a Function ofCompression-Ratio

In Phase I, signal reconstruction ability was quantified throughcorrelation coefficients, but the damage detection ability was evaluatedqualitatively through diagnostic images. In Phase II, Advent isdeveloping a methodology to generate probability-of-detection (POD)curves as a function of Compression Ratio (CR) to quantify accuracy ofdamage detection from undersampled data. Here, CR is defined as

${CR} = \frac{m \times k}{N \times P}$

where m is the number of temporally undersampled signal points, N is thefull signal length, k is the number of spatially undersampled sensors(or image pixels), and P is the number of sensors (or image pixels) in afully populated sensor array (or diagnostic image). Note that CR=1 forfully sampled data and less than 1 for undersampled data in both timeand space.

POD is a popular metric used to quantify the damage detection capabilityof NDE/SHM systems and is defined as the probability that a given damagewill be detected using a given inspection method. In practice, noNDE/SHM method provides 100% assurance on the damage size that ispossible to detect because there is statistical uncertainty in allmeasurements due to the number of specimens tested, operator experience,damage characteristics, structural material/geometry, environmentalchanges, etc. Therefore, NDE/SHM reliability is typically expressed interms of damage size that has a 90% POD with 95% confidence, aftertaking into account all variables that can affect detection.

In traditional sensing, as opposed to compressive sensing, using fewersensors or fewer sample points per measurement tends to shift the PODcurves to the right as shown in FIG. 34 . But as was found in the PhaseI study, the use of CS to reconstruct the undersampled data can restorethe damage detection capabilities, which will move the POD curves backtoward the fully sampled cases (FIG. 35 ).

In this methodology, POD curves will be generated to quantify the damagedetection capability as a function of CR. Rather than conducting afull-blown POD experimental study for different damage types anddifferent NDE/SHM methods, sensor data from previous POD studies will beused. Data from previous POD experimental studies on metals andcomposites will be downsampled and used as input into the softwarepackage. Damage detection results with various levels of CR will becompared with those from fully sampled experiments, and POD curves willbe generated.

Methodology to Quantify Benefits of Compressive Sensing for NDE/SHMApplications

Advent is developing a methodology to quantify the benefits that the CSapproach can provide for NDE/SHM of large structures. The savings willbe quantified in terms of: (1) data acquisition time, (2) data storagespace, (3) reduction in number of required sensors, and (4) total weightof NDE/SHM systems. These quantified savings will, in turn, be relatedto the POD curves developed above. The purpose is to give end users theability to make informed decisions for new applications regarding numberof sensors (or total NDE/SHM system weight) for a given critical damagesize and associated POD.

An example relationship between cost, compression ratio, and POD isshown in FIG. 36 . Here, cost could be data acquisition time, datastorage space, number of sensors, total NDE/SHM system weight, or acombination of all of the above. Each point on the curve corresponds toa particular CR, which corresponds to a particular POD curve.

Refined and Expanded Software Architecture with Example

The initial methodology for application to ultrasonic guided waves wasbroken down into four modules as shown in the diagram in FIG. 2 . Thepreprocessing modules are intended to be run only once in order togenerate the basis functions required for the Compressive Sensingalgorithms that are housed in the Stage 1 and Stage 2 modules. The Stage1 Signal Reconstruction module contains Compressive Sensing algorithmsto enable reconstruction of temporally undersampled sensor signals. TheStage 2 Path Reconstruction module contains Compressive Sensingalgorithms for reconstruction of spatially undersampled actuator-sensorpaths.

While the initial Compressive Sensing methodology was focused onreducing the data requirements for pitch-catch ultrasonic guided waves,the software architecture design allows it to be extended to otherNDE/SHM techniques. Advent has built upon the modular softwarearchitecture design and expanded it to include other NDE/SHM techniques.

Not all NDE/SHM methods will generate data that can be undersampled inboth time and space. Therefore, to generalize the architecture toaccommodate other types of NDE/SHM methods, the stages have beenredefined to be Stage 1 Temporal Reconstruction and Stage 2 SpatialReconstruction. Also, the NDE/SHM dataflow has been designed todistinguish between datasets that have been undersampled in both timeand space from those that have been undersampled in either time or space(FIG. 33 ). For example, data that has been undersampled in both spaceand time (e.g. the pitch-catch data) will flow through both Stage 1 andStage 2 as shown by the blue arrows in the figure. NDE data that hasbeen undersampled in time only (e.g. pulse-echo/A-scan or acousticemission data) will just flow through Stage 1 to reconstruct the signalsas shown by the red arrows. Similarly, data that has been undersampledin space only (e.g. C-scan data) will bypass Stage 1 and go directly toStage 2 to reconstruct the data for imaging as shown by the greenarrows.

The architecture and methodologies enable data reconstructioncapabilities for various NDE/SHM techniques, including pitch-catchultrasonic guided waves, pulse-echo (A-scan), B-scan, C-scan, Z-scan,acoustic emission, impact data, thermography, etc.

As before, the Preprocessing modules for Stage 1 and Stage 2 areintended to be run only once for each application. Their purpose is todetermine the best basis functions to use for a particular type ofNDE/SHM data to accurately reconstruct the data in time (Stage 1) and/orspace (Stage 2).

Example Using A-Scan and C-Scan Data

Previously, reconstruction of undersampled A-scan data and undersampledC-scan data was done separately, following the red and green arrows inFIG. 33 , respectively. The A-scan data is an ultrasonic pulse-echosignal (amplitude vs. time) from a single point, while the C-scan datais a 2-D image created by all the points where an A-scan signal wascollected (FIG. 37A). In other words, each A-scan signal represents onepixel in the C-scan image (FIG. 37B). This example follows the bluearrows in FIG. 33 by first reconstructing undersampled A-scan data toobtain pixel information, and then reconstructing a 2D C-scan image thathas missing pixels (missing A-scan data).

The data was obtained using USC's customized PVA TePLA SAM 300, whichhas an operating frequency range of 5-400 MHz. After defining thescanning areas and the central frequency, the machine discretizes thematerial domain into pixel points. Then an A-scan signal from everypixel point is acquired.

A comprehensive parametric study was conducted to generate C-scan imagesusing various combinations of compression ratios for the A-scan data(undersampled in time) and C-scan data (undersampled in space), and toquantify the reconstructions using correlation coefficients andprobability-of-detection. A composite plate was fabricated from two 254mm by 508 mm coupons, each with IM7/8552 unidirectional tape in a0/90-degree layup. The plate consists of a total of 40 plies(approximately 7.5 mm thick), with the top sheet having 24 plies[(0/90)_12] and the bottom sheet having 16 plies [(0/90)_8] as shown inFIG. 38 . The two coupons were bonded together using FM-300 adhesive.

For convenience, the plate was divided into 8 different regions, namingAAW, ABW, ACW, ADW, AAE, ABE, ACE, and ADE. Each of these regions werefurther subdivided into a 5×5 grid, with each grid measuringapproximately 25 mm×25 mm. Each grid segment can be identified by itsrow and column (e.g., row A and column 1).

Laser Bond Inspection (LBI) was applied at each grid point withdifferent incident fluence energies to induce localdebonds/delaminations of different sizes. Images of several sampledifferent grid regions and the applied incident fluence energies areshown in FIG. 39 . In general, the severity of the debonds/delaminationsincrease with the applied incident fluence energy.

A full 3D Z-scan was performed using SAM with a 25 MHz transducer, and aC-scan image was generated at the bondline (FIG. 40 ). Data from gridregion B7 was downsampled and then reconstructed using compressivesensing. The grid region is 125-pixels×125-pixels, for a total of 15,625pixels. Each pixel contains a time-domain signal vector that is 76,001sample points in length. These time-domain signals were initiallyreduced to 15,000 sample points by taking only the portion between thetop surface and bottom surface reflections. The 15,000-point vectorswere then randomly downsampled to various compression ratios. Initialanalysis was conducted using 750, 1500, 3000, and 3750 sample points forcompression ratios (CR) of 5%, 10%, 20%, and 25%, respectively. Theundersampled signals were then reconstructed using the Discrete CosineTransform as a basis function, and the images were regenerated as shownin FIG. 40 . The respective correlation coefficients (CC) were 0.218,0.415, 0.779, and 0.857. The LBI-induced debond/delamination can be seenat a CR of 10% and CC of 0.415 but is much clearer at a CR of 20% whenthe CC is above 0.7.

The above initial image reconstructions included low compression ratiosin the A-scan data, but were spatially fully sampled (all of the pixels)for the image reconstruction.

A comprehensive set of cases were then run for various combinations ofcompression ratios for both the A-scan data and C-scan/Z-scan data(randomly undersampled pixels).

FIG. 41 shows the resulting C-scan images as a function of A-scan andC-scan compression ratios. The horizontal axis corresponds to Stage 1compressive sensing (undersampled in time) and the vertical axiscorresponds to Stage 2 compressive sensing (undersampled in space). Thegreen area in the chart represents the zone with acceptablereconstructed C-scan images that have a correlation coefficient greaterthan 0.7. There are several reconstructed images to the left and belowthe green area that also have correlation coefficients above 0.7, but toguarantee good image reconstruction, it is advised to stay within thegreen region.

Methodology to Determine Probability-of-Detection as a Function ofCompression-Ratio

In Phase I, signal reconstruction ability was quantified throughcorrelation coefficients, but the damage detection ability was evaluatedqualitatively through diagnostic images. In Phase II, Advent isdeveloping a methodology to generate probability-of-detection (POD)curves as a function of Compression Ratio (CR) to quantify accuracy ofdamage detection from undersampled data. Here, CR is defined as:

${CR} = \frac{m \times k}{N \times P}$

where m is the number of temporally undersampled signal points, N is thefull signal length, k is the number of spatially undersampled sensors(or image pixels), and P is the number of sensors (or image pixels) in afully populated sensor array (or diagnostic image). Note that CR=1 forfully sampled data and less than 1 for undersampled data in both timeand space.

POD is a popular metric used to quantify the damage detection capabilityof NDE/SHM systems and is defined as the probability that a given damagewill be detected using a given inspection method. In practice, noNDE/SHM method provides 100% assurance on the damage size that ispossible to detect because there is statistical uncertainty in allmeasurements due to the number of specimens tested, operator experience,damage characteristics, structural material/geometry, environmentalchanges, etc. Therefore, NDE/SHM reliability is typically expressed interms of damage size that has a 90% POD with 95% confidence, aftertaking into account all variables that can affect detection.

In traditional sensing, as opposed to compressive sensing, using fewersensors or fewer sample points per measurement tends to shift the PODcurves to the right as shown in FIG. 42 . But as was found in the PhaseI study, the use of CS to reconstruct the undersampled data can restorethe damage detection capabilities, which will move the POD curves backtoward the fully sampled cases (FIG. 43 ).

Previously, signal reconstruction and diagnostic image reconstructionabilities have been quantified only through correlation coefficients.The quantification metrics have been expanded by developing amethodology to generate probability-of-detection (POD) curves as afunction of compression ratio (CR) and the subsequent correlationcoefficient (r) after reconstruction to quantify accuracy of damagedetection from undersampled data.

The objective is to demonstrate that using fewer sensors or fewer samplepoints per measurement (CR<100%) can result in the same, or nearly thesame, damage detection capabilities as fully sampled cases. Intraditional sensing, using fewer sensors or fewer sample points permeasurement tends to flatten and move the POD curves to the right. Butthe use of CS to reconstruct the undersampled data can restore thedamage detection capabilities, which will move the POD curves backtoward the fully-sampled cases.

There are many influencing factors that affect POD, including: theintrinsic capability of a given NDE/SHM method (physical process ofsignal detection of waves/rays from a material defect); damage type,size, and orientation; structural geometry and accessibility;environment (temperature, humidity, vibration); and human factors(inspector experience, inspection procedures, organizational protocol).Furthermore, there are many different methodologies/formulations forgenerating POD curves, depending on the application.

The goal of the developed methodology is to generalize the procedure forestimating the POD and make it applicable to all NDE/SHM methods thatuse CS to reconstruct undersampled data.

The methodology starts by representing the POD for the undersampled case(before CS reconstruction), POD_(u), as a function of the originalfully-sampled POD_(f).

POD _(u)(a)=POD _(f)(a′) where a′=C ₁(a−C ₂)

Here, a is the defect size, and a′ is a modified detectable defect sizedue to undersampling of the inspection data. C₁ represents a scaling ofthe detectable damage size, and changing the value will shift and skewthe POD curve. C₂ represents a constant jump in the detectable damagesize, and a change in value will result in a pure shift of the POD curveto the left or the right.

This generalized equation for a′ can be simplified by noting that bothundersampling in time (stage 1 CS) and undersampling in space (stage 2CS) will result in a scaling of the detectable damage size rather than aconstant jump. For example, for the case of the SAM data from thecomposite plate with induced debonds/delaminations (described inA-scan/C-scan composite plate example above) the CR is defined as:

${CR} = {\frac{m \times k}{N \times P} \times 100\%}$

where in is the number of temporally undersampled signal points in eachA-scan, N is the full signal length of each A-scan, k is the number ofspatially undersampled pixels in the C-scan, and P is the full number ofpixels in the C-scan. Note that CR=100% for fully sampled data and lessthan 100% for undersampled data in both time and space.

From geometry, undersampling in space results in a larger averagedistance between pixels, which effectively scales the detectable damagesize proportional to CR. Therefore, we can approximate a′ by settingC₁=CR and C₂=0, giving

POD _(u)(a)≈POD _(f)(CR×a)

Example POD curves for CR values of 25%, 20%, and 10% are shown in FIG.44 . These values correspond to the C-scan data from FIG. 41 , and thegenerated POD curves are before any CS reconstruction of the data.

Applying CS to reconstruct the undersampled data will restore the damagedetection capability. We measure the data reconstruction using thecorrelation coefficient, r. The higher the value of r, the better thereconstruction, and the more the POD curve will move back toward thefully-sampled case.

All NDE/SHM systems have an inherent noise level, and two consecutivefully-sampled measurements will not be identical to each other due tothe inherent noise in the system. The r value between two consecutivefully-sampled measurements is the highest r value we can expect a CSreconstruction to achieve, and if it does, then we can claim the data isfully reconstructed and the POD curve has shifted back to thefully-sampled case.

Using a similar formulation as before, the curves for the reconstructedcase, POD_(r), can be expressed as:

${{POD}_{r}(a)} \approx {{POD}_{f}\left( {\frac{r_{r}}{r_{f}} \times a} \right)}$

Here, POD, is the POD for the reconstructed data, POD_(f) is the POD forthe fully-sampled case, r_(r) is the correlation coefficient for thereconstructed case, and r_(f) is the correlation coefficient between twoconsecutive fully-sampled measurements.

Example POD curves of the C-scan data after CS reconstructions are shownin FIG. 45 . Here, we set r_(f)=0.95, and used r_(r)=0.857, 0.779, and0.415 (corresponding to the CR values of 25%, 20%, and 10%,respectively). As can be seen, the reconstructions of the data for theCR=25% and 20% cases nearly restore the POD to the original curve.

As another example, we use the image reconstructions from FIG. 41 thatshow the resulting C-scan images as a function of A-scan and C-scancompression ratios. The line in the chart in FIG. 41 separates theacceptable reconstructed C-scan images (correlation coefficients greaterthan 0.70) from the others. The combined compression ratio along thisline falls between 15% and 20%, and the POD along this line beforereconstruction is represented by the gray area in FIG. 46 . Afterreconstruction, the POD for cases along the line shift to the left andfall within the green area in FIG. 46 .

Methodology to Quantify Benefits of Compressive Sensing for NDE/SHMApplications

Advent has developed a methodology to quantify the benefits that the CSapproach can provide for NDE/SHM of large structures. The savings can bequantified in terms of: (1) data acquisition time, (2) data storagespace, (3) reduction in number of required sensors, and (4) total weightof NDE/SHM systems. These quantified savings, in turn, are related tothe POD curves developed above. The purpose is to give end users theability to make informed decisions for new applications regarding numberof sensors (or total NDE/SHM system weight) for a given critical damagesize and associated POD.

A general example relationship between cost, compression ratio, and PODis shown in FIG. 47 . Here, cost could be data acquisition time, datastorage space, number of sensors, total NDE/SHM system weight, or acombination of all the above. Each point on the curve corresponds to aparticular CR, which corresponds to a particular POD curve.

An example relationship between data storage space (cost), compressionratio, and POD is shown in FIG. 48 . Recall from the composite plateexample above that each C-scan image contains 125-pixels×125-pixels, fora total of 15,625 pixels. And each pixel contains a time-domain signalvector that is 76,001 sample points in length. These time-domain signalswere initially reduced to 15,000 sample points by taking only theportion between the top surface and bottom surface reflections. This byitself reduced the file size to less than 20% of its original size. Thenthe data file size was reduced further using random downsampling andcompressive sensing. The chart in FIG. 48 is showing the furtherreduction that is only due to compressive sensing (which is in additionto the already-compressed file size).

As can be seen, the system maintains high POD capability, even when thefile size is reduced to about 10% of the already-compressed file size(or about 2% of the original file size).

The methods and systems disclosed herein may be carried out on one ormore electronic computing platforms such as a hand-held device, laptop,desktop computer, workstation with a single or multi-core processor,server with multiple processors, and/or cluster of computers, forpurposes of example only and not intended to be limiting. For example, adiagnostic device may include a processor and server in communicationwith a sensor array. The diagnostic device may provide a diagnosticdisplay screen to a user and may communicate commands to the server forfurther processing of the user's inputs through the diagnostic screen,such as changing the display, engaging software platforms, engagingsubroutines, switching modules, etc.

According to another aspect, computer readable media and devices areprovided to collect signals, analyze and reconstruct same and providedamage diagnostics for a structure according to the various processesdescribed herein. Many of the techniques described herein can beimplemented in hardware, firmware, software, or a combination thereof.In one embodiment, the technology is on a programmable computer thatincludes a processor, a processor-readable storage medium (includingvolatile and non-volatile memory and/or storage elements), and suitableinput/output devices, respectively. Implemented in a computer programthat runs. The program code is applied to the data input using the inputdevice to perform the functions described and to generate outputinformation, such as reconstructed signals and/or damageestimates/projections. The output information applies to one or moreoutput devices. Also, each program is preferably implemented in ahigh-level procedure or object-oriented programming language tocommunicate with a computer system. However, the program can beimplemented in assembly or machine language if desired. In any case, thelanguage can be a compiled or interpreted language.

Various modifications and variations of the described methods,pharmaceutical compositions, and kits of the disclosure will be apparentto those skilled in the art without departing from the scope and spiritof the disclosure. Although the disclosure has been described inconnection with specific embodiments, it will be understood that it iscapable of further modifications and that the disclosure as claimedshould not be unduly limited to such specific embodiments. Indeed,various modifications of the described modes for carrying out thedisclosure that are obvious to those skilled in the art are intended tobe within the scope of the disclosure. This application is intended tocover any variations, uses, or adaptations of the disclosure following,in general, the principles of the disclosure and including suchdepartures from the present disclosure come within known customarypractice within the art to which the disclosure pertains and may beapplied to the essential features herein before set forth.

1.-19. (canceled)
 20. A method for employing compressive sensing tosense damage in a structure comprising: employing at least one spatialsensor array; obtaining at least one temporal sensor signal from the atleast one spatial sensor array; wherein either the at least one spatialsensor array is undersampled or the at least one temporal signal isundersampled but not both; employing compressive sensing to reconstructdata from the at least one temporally undersampled sensor signal togenerate a reconstructed temporal signal, if the at least one temporalsignal is undersampled; or employing compressive sensing to reconstructdata from the at least one spatial sensor array that is undersampled togenerate reconstructed spatial data, if the at least one spatial sensorarray is undersampled; and generating a diagnostic image of thestructure based on either the reconstructed temporal signal or thereconstructed spatial data.
 21. The method for employing compressivesensing to sense damage in a structure of claim 20, wherein the at leastone spatially undersampled sensor data results from missingactuator-sensor paths in the sensor array.
 22. The method for employingcompressive sensing to sense damage in a structure of claim 20, whereinthe method is employed to diagnose a space structure.
 23. The method foremploying compressive sensing to sense damage in a structure of claim20, wherein the method employs at least one algorithm processed by anultrasound software package.
 24. The method for employing compressivesensing to sense damage in a structure of claim 20, wherein the methodreconstructs at least one sensor signal via using the at least onetemporally undersampled signal, an appropriate basis function, and ameasurement matrix.
 25. The method for employing compressive sensing tosense damage in a structure of claim 24, further comprising utilizing asubroutine to determine a best basis function for the at least onetemporally undersampled signal.
 26. The method for employing compressivesensing to sense damage in a structure of claim 25, further comprisingwherein the subroutine incorporate a Gini Index to find a basis functionthat provides a sparsest signal in a transform domain.
 27. The methodfor employing compressive sensing to sense damage in a structure ofclaim 20, wherein the at least one temporally undersampled signalcomprises an ultrasound signal.
 28. The method for employing compressivesensing to sense damage in a structure of claim 23, wherein thealgorithm is a sparse recovery algorithm.
 29. The method for employingcompressive sensing to sense damage in a structure of claim 28, whereinthe sparse recovery algorithm comprises

1-norm minimization.
 30. The method for employing compressive sensing tosense damage in a structure of claim 20, wherein the at least onespatial sensor array is not affixed to the structure.
 31. A diagnosticmethod for employing compressive sensing comprising: receiving a sensorsignal from a sensor network not affixed to a structure that transmitsat least one signal to the structure and receives the at least onesignal after encountering the structure; wherein when the sensor datareceived from the sensor network is undersampled in time or undersampledin space, but not both, the diagnostic method: engage at least onepreprocessing module to run only once to generate at least one basisfunction for at least one compressive sensing algorithm; engage at leastone signal reconstruction module containing the at least one compressivesensing algorithm, wherein the at least one reconstruction modulereconstructs at least one temporally undersampled sensor signal; orengage at least one second reconstruction module containing the at leastone compressive sensing algorithm to reconstruct at least one spatiallyundersampled sensor data; and determining whether damage has occurred tothe structure via reconstruction of the at least one temporallyundersampled sensor signal or the at least one spatially undersampledsensor data.
 32. The diagnostic method for employing compressive sensingof claim 31, wherein the diagnostic method is employed with at least onestructural health monitoring system.
 33. The diagnostic method foremploying compressive sensing of claim 31, wherein the diagnostic methodengage at least two preprocessing modules.
 34. The diagnostic method foremploying compressive sensing of claim 31, wherein the at least onesignal is an ultrasound signal.
 35. The diagnostic method for employingcompressive sensing of claim 31, wherein the method reconstructs atleast one reconstructed sensor signal via using the at least one signal,an appropriate basis function, and a measurement matrix.
 36. Thediagnostic method for employing compressive sensing of claim 35, furthercomprising utilizing a subroutine to determine a best basis function forthe at least one signal.
 37. The diagnostic method for employingcompressive sensing of claim 36, further comprising wherein thesubroutine incorporate a Gini Index to find a basis function thatprovides a sparsest signal in a transform domain.
 38. The diagnosticmethod for employing compressive sensing of claim 31, wherein at leastone compressive sensing algorithm is a sparse recovery algorithm. 39.The diagnostic method for employing compressive sensing of claim 38,wherein the sparse recovery algorithm comprises

1-norm minimization.